Gödel, Escher, Bach: An Eternal Golden Braid
by Douglas Hofstadter
My favorite book of all time is The Brothers Karamazov. I believe it covers almost every aspect of the human condition with unparalleled nuance and insight. Gödel, Escher, Bach: An Eternal Golden Braid is the nonfiction equivalent of Dosteyevsky’s magnum opus.
Alternating chapters between a pleasantly conversational textbook and a dialogue between a fictional cast of characters including Achilles and the Tortoise (of Zeno’s Paradox‘s fame), Hofstadter’s Pulitzer-winning work travels from the theoretical foundation of information and meaning to it’s implications regarding Artificial Intelligence and consciousness. In spite of the fact that this book is more than three decades old, it is still a masterful synthesis of the arguments that now seem mainstream in academic and intellectual communities (particularly telling of it’s age is the repeated references to chess playing machines, which at the time, could not come close to besting a skilled chess player). I only wish I could have been alive when the book was first released to appreciate the bravado and radical nature of the arguments in its own time.
At the heart of the book is Hofstadter’s favorite subject: recursive loops. He explores the power of loops in number theory, in painting, and in music. What makes GEB:EGB stand out is how adroitly woven together the three topics are. Using M.C. Escher’s famous impossible drawings as a starting point, Hofstadter explores Gödel’s incompleteness theorems that revolutionized mathematics and logic, all the while writing chapters in a format that directly mirrors select Bachian canons. It is a work of art that discusses the relationship between logic and painting while describing the creation of complex systems from simple parts, written as an homage to thematically relevant classical music. It does not get more wonderfully complicated than this.
But my use of “wonderfully” may be idiosyncratic to my tastes. This book is VERY far-reaching. With so many tangents and nifty ideas that one may fix upon, as well as direct challenges and puzzles to the reader from the author, sticking to the central theme of the book can be difficult. But if you enjoy a sort of academic, “Follow-the-Leader” type experience, taking you in seemingly random connections that, on faith, you believe have a payoff, then read on.
Hofstadter introduces the concepts of Gödel numbering, which transforms logical statements into natural numbers, and then demonstrates how Gödel used this system to rip a gaping hole in the prevailing number theory that, at the time, was hoped to be perfect. By forcing the system to reference itself, Gödel was able to show that there must be true statements that the system cannot prove. Put more directly, in any “well-formed” number theory (call it X), number theory X can be used to create a proof that number theory X has self-consistency only if number theory X is inconsistent. And vice versa. The end result being that there is no all-encompassing number theory, because any theory cannot fully encompass all statements about itself. To quote from the book: “The paraphrase of Gödel’s Theorem says that for any record player, there are records which it cannot play because they will cause its indirect self-destruction.”
Hofstadter uses Gödel’s breakthrough of an inescapable hole, or a necessary context, to further discuss recursive systems and looped hierarchies, showing that circular hierarchies are only possible within the context of a larger, all-encompassing system. This idea is demonstrated in some of Escher’s work, most prominently in his “Print Gallery,” which features a man, staring at a painting, which is of a town, which includes a gallery, which houses a man, staring at a painting…etc. In Gödel’s painting, there is a blank spot left in the center, used to sign the artist’s name, but also included because of the necessary of it. Hofstadter points out that if all the perspectives had been continued, the joining of the picture would shatter it’s consistency. The picture, like number theory, only works if there was a hole in it.
That hole, according to Hofstadter, is a shared context. And while there exists complex systems that have a whole greater than the sum of its parts (ant colonies, bee hives, brains), the basic parts upon which the entire system is built (ants, bees, neurons) must obey a system-encompassing context (such as laws of physics), no matter how recursive or self-modifying that system is.
This is but one example of the rewarding challenges of the GEB. It’s a book that takes far flung concepts and weaves them together beautifully but at times opaquely, something that I will undoubtedly learn again from after rereading. There are so many layers to this book, meta doesn’t even begin to describe it. I consider it one of the most brilliant pieces of art and intellect I have ever experienced, and if you have many spare hours to read, and even more to spend pondering, I cannot recommend this book enough. If, as Hofstadter puts it, “you enjoy thinking about thinking about thinking.”